5. the graph of a function f is shown in the figure above. which of the following statements must be false…

Explanation:

Step1: Analyze option A

The function graph is shown on the interval $[0, b]$, so $f(x)$ is defined for $0\leq x\leq b$.

Step2: Analyze option B

There is a corresponding - point on the graph at $x = b$, so $f(b)$ exists.

Step3: Analyze option C

At $x = b$, the function has a sharp - turn. The left - hand and right - hand derivatives at $x = b$ are not equal. According to the definition of the derivative $f^{\prime}(x)=\lim_{\Delta x\rightarrow0}\frac{f(x + \Delta x)-f(x)}{\Delta x}$, for a function to be differentiable at a point, the left - hand and right - hand limits of the difference quotient must be equal. Since there is a sharp - turn at $x = b$, $f^{\prime}(b)$ does not exist.

Step4: Analyze option D

As $x\rightarrow a^{-}$, the function is smooth on the left - hand side of $x = a$, so $\lim_{x\rightarrow a^{-}}f^{\prime}(x)$ exists.

Answer:

C. $f^{\prime}(b)$ exists.